Session #2: Homework Solutions

Problem #1

In all likelihood, the Soviet Union and the United States together in the past exploded about ten devices underground per year.

(a) If each explosion converted about 10 g of matter into an amount of energy (a conservative estimate), how many kJ of energy were released per device?

(b) If the energy of these ten devices had been used for propulsion to substitute for gasoline , how many gallons of gasoline would not have had to be burned per year? (One gallon of gasoline releases about 1.5 x 105 kJ during combustion.)

Solution

Required: ΔΔE = mc2

1 kg (a) ΔΔE = mc28 = 10 g x x (3 x 10 ms-1 )2 1000 g

= 9 x 1014 kg m 2 s− 2 = 9 x 1014 J = 9 x 1011 kJ/bomb

11 12 (b) Etotal = 10 x 9 x 10 = 9 x 10 kJ/year

1 gal No. gallons of gasoline saved = x 9 x 1012 kJ/year 1.5 x 105 kJ

= 6 x 107 gallons/year

Problem #2

How much (in kg) is required to completely convert 1 mole of C2H6 into CO2 and H2O?

Solution

To get the requested answer, let us formulate a “stoichiometric” equation (molar quantities) for the reaction: C26 H + 70 → 2CO2 + 3H 2 O. Each C2H6 (ethane) requires 7 oxygen for complete combustion. In molar quantities: 1 mole of C2H6 = 2 x 12.01 + 6 x 1.008 = 30.07 g requires

7 x 15.9984 g = 1.12 x 102 oxygen = 0.112 kg oxygen

We recognize the oxygen forms , O2, and therefore a more appropriate formulation would be: C26 H + 7/2 O2 → 2CO2 + 3H 2 O. The result would be the same.

Problem # 3

A nucleus of number 56 contains 30 neutrons. An “” of this element has 23 . Write the symbol of this ion and give the ionic charge as a superscript on the right.

Solution

56 +++ 56 +++ 26 A = 26Fe

Problem # 4

Magnesium (Mg) has the following isotopic distribution:

24Mg 23.985 amu at 0.7870 fractional abundance

25Mg 24.986 amu at 0.1013 fractional abundance

26Mg 25.983 amu at 0.1117 fractional abundance

What is the atomic weight of magnesium (Mg) according to these data?

Solution

The atomic weight is the arithmetic average of the atomic weights of the , taking into account the fractional abundance of each .

23.985 x 0.7870 + 24.986 x 0.1013 + 25.983 x 0.1117 At.Wt. = = 24.310 0.7870 + 0.1013 +0.1117

Problem # 5

(a) Balance the equation for the reaction between CO and O2 to form CO2.

(b) If 32.0 g of oxygen react with CO to form dioxide (CO2), how much CO was consumed in this reaction?

Solution

(a) CO + 1/2 O22 → CO

(b) [Information only at 1 digit!] Molecular Weight (M.W.) of O2: 32.0 (M.W.) of CO: 28.0

available oxygen: 32.0g = 1 mole, correspondingly the reaction involves 2 moles of CO [see (a)]:

O22 + 2 CO → 2 CO

mass of CO reacted = 2 moles x 28 g /mole= 56.0 g

Problem # 6

One mole of electromagnetic radiation (light, consisting of energy packages called photons) has an energy of 171 kJ/mole photons. (a) Determine the wavelength of this light and its position in the visible spectrum.

(b) Determine the frequency of this radiation (in SI units).

Solution

(We know: Ephoton = hν = hc/λ to determine the wavelength associated with a photon we need to know its energy).

171kJ 1.71 x 105 J 1 mole (a) E = = x mole mole 6.02 x 1023 photons

2.84 x 10-19 J hc = ; E = 2.84 x 10-19 J = hν = photon photon λ

m 6.63 x 10-34 Js x 3 x 108 hc λ = = s = 7.00 x 10-7 m -19 Ephoton 2.84 x 10 J

= 700 nm (red light) (b) (IS or SI units are in m, k, s)

λν = c

m 3 x 108 c ν = = s = 4.29 x 1014 s− 1 = 4.29 x 1014 Hz λ 7.00 x 10-7 m

Problem # 7

Determine the velocity of an (in m/s) that has been subjected to an accelerating potential V of 150 . (The energy imparted to an electron by an accelerating potential of one Volt is 1.6 x 10–19 ; shows that the dimensions of charge x potential correspond to those of energy; thus: 1 electron Volt (1eV) = 1.6 x 10–19 x 1 Volt = 1.6 x 10–19 Joules.)

Solution

2 We know: Ekin = mv / 2 = e x V (charge applied potential) -31 me = 9.1 x 10 kg

2 Ekin = e x V = mv/2

2eV 2 x 1.6 x 10-19 x 150 v == = 7.26 x 106 m/s m 9.1 x 10-31

Problem # 8

Determine in units of eV the energy of a photon (hν ) with the wavelength of 800 nm.

Solution m 6.63 x 10-34 [Js] x 3 x 108 [ ] hc l eV s l eV E(eV) = x = x λ 1.6 x 10-19 J 8.00 x 10-7 m 1.6 x 10-19 J

= 1.55

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3.091SC Introduction to Solid State Fall 2009